The current situation is like this:
The pool passed to open is a session pool. Things allocated there will
live during the session. The pools passed to the other functions are used
for the work of those functions. But those functions sometimes need to
allocate things in the session pool. An example is the authenticated
username. Well, your point is that the docstring is unclear and I aggree.

I am accumulating such changes for some days. I am pretty sure I will get
some trivial things from your review. This big wrapper and deprecation
made my head spin somewhat and I probably missed some reindentation and
such.